Problem 48 - Algebra 2

Question

Assume that \(\log 4 \approx 0.6021, \log 5 \approx 0.6990,\) and \(\log 6 \approx 0.7782 .\) Use the properties of logarithms to evaluate each expression. Do not use your calculator. \(\log 1.5\)

Step-by-Step Solution

Answer

\(\log(1.5) \approx 0.1761\)

Step 1 - Express the given log value using known log values

Recognize that \(1.5\) can be expressed as \(\frac{6}{4}\) or \(\frac{3}{2}\). To use the given logarithm values, express \(1.5\) as \(\frac{6}{4}\), because the log values for \(4\) and \(6\) are known.

Step 2 - Apply the Quotient Rule of Logarithms

The Quotient Rule states that \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\). Apply this to \(\log(1.5)\) or \(\log\left(\frac{6}{4}\right)\), resulting in \(\log(6) - \log(4)\).

Step 3 - Substitute the known log values

Substitute the given values for \(\log(6)\) and \(\log(4)\) into the expression: \[\log(1.5) = \log\left(\frac{6}{4}\right) = \log(6) - \log(4) \approx 0.7782 - 0.6021.\]

Step 4 - Perform the subtraction

Subtract the log values: \[0.7782 - 0.6021 = 0.1761.\] This is the value of \(\log(1.5)\).

Key Concepts

Quotient Rule of Logarithms

Understanding how to evaluate logarithmic expressions without a calculator involves getting familiar with certain rules and properties. One of the quintessential rules is the Quotient Rule of Logarithms. It states that the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. In mathematical terms, this is expressed as:
\[\begin{equation}\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\end{equation}\]
This rule is incredibly useful when you're dealing with division inside a logarithm. For instance, if you needed to find the value of \(\log(1.5)\), and you know the values for \(\log(4)\) and \(\log(6)\), you can express 1.5 as \(\frac{6}{4}\), then apply the Quotient Rule to find the solution as shown in the step by step solution of the exercise.
For learners aiming to master logarithms, internalizing the quotient rule is a must. It simplifies complex expressions and enables you to work with more familiar logarithmic values.

Logarithmic Expressions

A logarithmic expression is a way of writing exponents in the form of logs. Logarithms can often seem intimidating, but once you break them down, they're quite digestible. They are the inverse of exponential functions, meaning they undo the exponentiation process. Taking the example of \(\log(1.5)=\log\left(\frac{6}{4}\right)\), the goal is to rewrite the expression in such a way that it utilizes known values or simplifies to a point where it can be evaluated.
In the practice of evaluating logarithms, you'll encounter different forms of expressions like products, quotients, or powers. Familiarity with the properties of logarithms assists in rewriting these complex forms into simpler terms—for example, using the Quotient Rule to break down a fraction, as seen before.

Converting Logarithmic Expressions

In our case with \(\log(1.5)\), we converted the decimal to a fraction with familiar logarithmic values before applying the Quotient Rule. These conversion steps are often what make or break your ability to calculate a logarithm without a calculator. Therefore, it's crucial to practice rewriting and recognizing different forms that logarithmic expressions can take.

Properties of Logarithms

Apart from the Quotient Rule, there are other fundamental properties of logarithms that are equally important. The three main properties that often come into play are the Product Rule, the Power Rule, and the aforementioned Quotient Rule. Each of these is a tool that, when used correctly, simplifies complex logarithmic expressions into more manageable pieces.

• The Product Rule states that the log of a product is equal to the sum of the logs of the factors: \(\log(ab) = \log(a) + \log(b)\).
• The Power Rule states that the log of a power is the exponent times the log of the base: \(\log(a^b) = b\cdot\log(a)\).
• The Quotient Rule, as discussed, states that the log of a quotient is the difference of the logs: \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\).

By applying these properties, you can break down and evaluate logs without resorting to a calculator. Mastering these properties also provides a foundation for further topics in mathematics such as solving logarithmic equations and analyzing logarithmic functions. They are not just rules to remember but tools to understand the nature of logarithmic relationships.
In the given problem, we primarily used the Quotient Rule because of the fractional form of 1.5, but knowing when and how to use each property is essential for tackling a variety of logarithmic challenges.